##
**Mathematical models and finite elements for reservoir simulation. Single phase, multiphase and multicomponent flows through porous media.**
*(English)*
Zbl 0603.76101

Studies in Mathematics and its Applications, Vol. 17. Amsterdam etc.: North-Holland. XI, 376 p. $ 80.00; Dfl. 200.00 (1986).

This book provides a synthetic presentation of the models governed by the Muskat (relative permeability) law and which are used in reservoir simulation, the fractured case being included. This is mainly achieved by the utilization of the new concept of global (or reduced) pressure, which enables the formulation of all the specific problems by one pressure equation coupled with one or several saturation or concentration equations.

For the immiscible flow models, a rigorous study is developed, this material being perhaps today’s most comprehensive mathematical treatment of the two-phase equations, as it takes into account the largest number of relevant physical properties.

The volume deals also with the relation between various models of fluid flows through porous media. Existence and uniqueness theorems for slightly compressible monophasic fields are proved. Generalizations of the global pressure transformation are presented for models of three- phase compressible flows, black-oil and compositional, miscible displacements.

In the last part, written by the second author, a finite element approximation technique is used for the two-phase incompressible flow. The method is concerned with mixed finite elements for the pressure equation and upstream weighted discontinuous finite elements with slope limiters for the saturation equation.

This book addresses to applied mathematicians and graduate students as an introduction to the reservoir simulation problems. Last but not least, it will surely be of interest for the researchers involved in oil recovery simulation.

For the immiscible flow models, a rigorous study is developed, this material being perhaps today’s most comprehensive mathematical treatment of the two-phase equations, as it takes into account the largest number of relevant physical properties.

The volume deals also with the relation between various models of fluid flows through porous media. Existence and uniqueness theorems for slightly compressible monophasic fields are proved. Generalizations of the global pressure transformation are presented for models of three- phase compressible flows, black-oil and compositional, miscible displacements.

In the last part, written by the second author, a finite element approximation technique is used for the two-phase incompressible flow. The method is concerned with mixed finite elements for the pressure equation and upstream weighted discontinuous finite elements with slope limiters for the saturation equation.

This book addresses to applied mathematicians and graduate students as an introduction to the reservoir simulation problems. Last but not least, it will surely be of interest for the researchers involved in oil recovery simulation.

Reviewer: D.Poliševski

### MSC:

76T99 | Multiphase and multicomponent flows |

76V05 | Reaction effects in flows |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76N20 | Boundary-layer theory for compressible fluids and gas dynamics |

76N15 | Gas dynamics (general theory) |

46N99 | Miscellaneous applications of functional analysis |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

76S05 | Flows in porous media; filtration; seepage |